Further inequalities for the (generalized) Wills functional

TL;DR

This paper explores new properties and inequalities of the Wills functional for convex bodies, including bounds, geometric inequalities, and extremal shapes, using integral and log-concave function representations.

Contribution

It introduces novel bounds and inequalities for the Wills functional, extending its analysis through integral and log-concave function techniques in a more general setting.

Findings

Upper bounds for the Wills functional in terms of volume

Brunn-Minkowski and Rogers-Shephard type inequalities established

The cube of edge-length 2 maximizes the Wills functional among symmetric convex bodies

Abstract

The Wills functional $\mathcal{W}(K)$ of a convex body $K$, defined as the sum of its intrinsic volumes $\mathrm{V}_i(K)$, turns out to have many interesting applications and properties. In this paper we make profit of the fact that it can be represented as the integral of a log-concave function, which, furthermore, is the Asplund product of other two log-concave functions, and obtain new properties of the Wills functional (indeed, we will work in a more general setting). Among others, we get upper bounds for $\mathcal{W}(K)$ in terms of the volume of $K$, as well as Brunn-Minkowski and Rogers-Shephard type inequalities for this functional. We also show that the cube of edge-length 2 maximizes $\mathcal{W}(K)$ among all $0$-symmetric convex bodies in John position, and we reprove the well-known McMullen inequality $\mathcal{W}(K)\leq e^{\mathrm{V}_1(K)}$ using a different approach.